Average Error: 32.3 → 13.0
Time: 2.9m
Precision: 64
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]
↓
\[\frac{\frac{\sqrt[3]{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}{\frac{\tan k}{\frac{\frac{1}{{\left(\sqrt[3]{t}\right)}^{3}}}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}} \cdot \frac{\frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{{\left(\sqrt[3]{t}\right)}^{3}}}{1}}{\frac{1}{\frac{\ell}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}}\]
\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}
↓
\frac{\frac{\sqrt[3]{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}{\frac{\tan k}{\frac{\frac{1}{{\left(\sqrt[3]{t}\right)}^{3}}}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}} \cdot \frac{\frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{{\left(\sqrt[3]{t}\right)}^{3}}}{1}}{\frac{1}{\frac{\ell}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}}
double f(double t, double l, double k) {
        double r466417 = 2.0;
        double r466418 = t;
        double r466419 = 3.0;
        double r466420 = pow(r466418, r466419);
        double r466421 = l;
        double r466422 = r466421 * r466421;
        double r466423 = r466420 / r466422;
        double r466424 = k;
        double r466425 = sin(r466424);
        double r466426 = r466423 * r466425;
        double r466427 = tan(r466424);
        double r466428 = r466426 * r466427;
        double r466429 = 1.0;
        double r466430 = r466424 / r466418;
        double r466431 = pow(r466430, r466417);
        double r466432 = r466429 + r466431;
        double r466433 = r466432 + r466429;
        double r466434 = r466428 * r466433;
        double r466435 = r466417 / r466434;
        return r466435;
}
↓
double f(double t, double l, double k) {
        double r466436 = 2.0;
        double r466437 = cbrt(r466436);
        double r466438 = t;
        double r466439 = cbrt(r466438);
        double r466440 = 3.0;
        double r466441 = pow(r466439, r466440);
        double r466442 = l;
        double r466443 = r466441 / r466442;
        double r466444 = k;
        double r466445 = sin(r466444);
        double r466446 = r466443 * r466445;
        double r466447 = r466437 / r466446;
        double r466448 = tan(r466444);
        double r466449 = 1.0;
        double r466450 = r466449 / r466441;
        double r466451 = 2.0;
        double r466452 = 1.0;
        double r466453 = r466444 / r466438;
        double r466454 = pow(r466453, r466436);
        double r466455 = fma(r466451, r466452, r466454);
        double r466456 = sqrt(r466455);
        double r466457 = r466450 / r466456;
        double r466458 = r466448 / r466457;
        double r466459 = r466447 / r466458;
        double r466460 = r466437 * r466437;
        double r466461 = r466460 / r466441;
        double r466462 = r466461 / r466449;
        double r466463 = r466442 / r466456;
        double r466464 = r466449 / r466463;
        double r466465 = r466462 / r466464;
        double r466466 = r466459 * r466465;
        return r466466;
}

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

  1. Initial program 32.3

    \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)}\]

  2. Simplified32.4

    \[\leadsto \color{blue}{\frac{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}}\]
  3. Using strategy rm

  4. Applied add-cube-cbrt32.5

    \[\leadsto \frac{\frac{\frac{2}{\frac{{\color{blue}{\left(\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}\right)}}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\]

  5. Applied unpow-prod-down32.5

    \[\leadsto \frac{\frac{\frac{2}{\frac{\color{blue}{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}}}{\ell \cdot \ell} \cdot \sin k}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\]

  6. Applied times-frac25.8

    \[\leadsto \frac{\frac{\frac{2}{\color{blue}{\left(\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell}\right)} \cdot \sin k}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\]

  7. Applied associate-*l*24.0

    \[\leadsto \frac{\frac{\frac{2}{\color{blue}{\frac{{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right)}^{3}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\]
  8. Using strategy rm

  9. Applied unpow-prod-down24.0

    \[\leadsto \frac{\frac{\frac{2}{\frac{\color{blue}{{\left(\sqrt[3]{t}\right)}^{3} \cdot {\left(\sqrt[3]{t}\right)}^{3}}}{\ell} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\]

  10. Applied associate-/l*18.0

    \[\leadsto \frac{\frac{\frac{2}{\color{blue}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\]
  11. Using strategy rm

  12. Applied *-un-lft-identity18.0

    \[\leadsto \frac{\frac{\frac{2}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{\color{blue}{1 \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}{\tan k}\]

  13. Applied associate-*l/17.1

    \[\leadsto \frac{\frac{\frac{2}{\color{blue}{\frac{{\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}}}{1 \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\]

  14. Applied associate-/r/16.9

    \[\leadsto \frac{\frac{\color{blue}{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)} \cdot \frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}}{1 \cdot \mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}{\tan k}\]

  15. Applied times-frac15.3

    \[\leadsto \frac{\color{blue}{\frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{1} \cdot \frac{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}{\tan k}\]

  16. Applied associate-/l*13.2

    \[\leadsto \color{blue}{\frac{\frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{1}}{\frac{\tan k}{\frac{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}}\]
  17. Using strategy rm

  18. Applied add-sqr-sqrt13.3

    \[\leadsto \frac{\frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{1}}{\frac{\tan k}{\frac{\frac{\ell}{{\left(\sqrt[3]{t}\right)}^{3}}}{\color{blue}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}}}\]

  19. Applied div-inv13.3

    \[\leadsto \frac{\frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{1}}{\frac{\tan k}{\frac{\color{blue}{\ell \cdot \frac{1}{{\left(\sqrt[3]{t}\right)}^{3}}}}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}}\]

  20. Applied times-frac13.6

    \[\leadsto \frac{\frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{1}}{\frac{\tan k}{\color{blue}{\frac{\ell}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}} \cdot \frac{\frac{1}{{\left(\sqrt[3]{t}\right)}^{3}}}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}}}\]

  21. Applied *-un-lft-identity13.6

    \[\leadsto \frac{\frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{1}}{\frac{\color{blue}{1 \cdot \tan k}}{\frac{\ell}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}} \cdot \frac{\frac{1}{{\left(\sqrt[3]{t}\right)}^{3}}}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}}\]

  22. Applied times-frac13.8

    \[\leadsto \frac{\frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{1}}{\color{blue}{\frac{1}{\frac{\ell}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}} \cdot \frac{\tan k}{\frac{\frac{1}{{\left(\sqrt[3]{t}\right)}^{3}}}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}}}\]

  23. Applied add-cube-cbrt13.8

    \[\leadsto \frac{\frac{\frac{2}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}}{\frac{1}{\frac{\ell}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}} \cdot \frac{\tan k}{\frac{\frac{1}{{\left(\sqrt[3]{t}\right)}^{3}}}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}}\]

  24. Applied add-cube-cbrt13.8

    \[\leadsto \frac{\frac{\frac{\color{blue}{\left(\sqrt[3]{2} \cdot \sqrt[3]{2}\right) \cdot \sqrt[3]{2}}}{{\left(\sqrt[3]{t}\right)}^{3} \cdot \left(\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k\right)}}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{1}{\frac{\ell}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}} \cdot \frac{\tan k}{\frac{\frac{1}{{\left(\sqrt[3]{t}\right)}^{3}}}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}}\]

  25. Applied times-frac13.8

    \[\leadsto \frac{\frac{\color{blue}{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{{\left(\sqrt[3]{t}\right)}^{3}} \cdot \frac{\sqrt[3]{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\frac{1}{\frac{\ell}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}} \cdot \frac{\tan k}{\frac{\frac{1}{{\left(\sqrt[3]{t}\right)}^{3}}}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}}\]

  26. Applied times-frac13.8

    \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{{\left(\sqrt[3]{t}\right)}^{3}}}{\sqrt[3]{1} \cdot \sqrt[3]{1}} \cdot \frac{\frac{\sqrt[3]{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}{\sqrt[3]{1}}}}{\frac{1}{\frac{\ell}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}} \cdot \frac{\tan k}{\frac{\frac{1}{{\left(\sqrt[3]{t}\right)}^{3}}}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}}\]

  27. Applied times-frac13.0

    \[\leadsto \color{blue}{\frac{\frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{{\left(\sqrt[3]{t}\right)}^{3}}}{\sqrt[3]{1} \cdot \sqrt[3]{1}}}{\frac{1}{\frac{\ell}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}} \cdot \frac{\frac{\frac{\sqrt[3]{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}{\sqrt[3]{1}}}{\frac{\tan k}{\frac{\frac{1}{{\left(\sqrt[3]{t}\right)}^{3}}}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}}}\]

  28. Final simplification13.0

    \[\leadsto \frac{\frac{\sqrt[3]{2}}{\frac{{\left(\sqrt[3]{t}\right)}^{3}}{\ell} \cdot \sin k}}{\frac{\tan k}{\frac{\frac{1}{{\left(\sqrt[3]{t}\right)}^{3}}}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}} \cdot \frac{\frac{\frac{\sqrt[3]{2} \cdot \sqrt[3]{2}}{{\left(\sqrt[3]{t}\right)}^{3}}}{1}}{\frac{1}{\frac{\ell}{\sqrt{\mathsf{fma}\left(2, 1, {\left(\frac{k}{t}\right)}^{2}\right)}}}}\]

Reproduce

herbie shell --seed 2019305 +o rules:numerics
(FPCore (t l k)
  :name "Toniolo and Linder, Equation (10+)"
  :precision binary64
  (/ 2 (* (* (* (/ (pow t 3) (* l l)) (sin k)) (tan k)) (+ (+ 1 (pow (/ k t) 2)) 1))))